Algorithm Design

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Chapter 4 Greedy Algorithms
leaf v, and ff u is the parent of v in T, then h(u) >_ h(v). We place each point
in P at a distinct leaf in T. Now, for any pair of points p~ and Pi, their
distance ~(p~, Pi) is defined as follows. We determine the least common
ancestor v in T of the leaves containing p~ and Pi, and define ~(p~,
We say that a hierarchical metric r is consistent with our distance
function d if, for all pairs i,j, we have r(p~,pl) _< d(p~,Pi).
Give a polynomial-time algorithm that takes the distance function d
and produces a hierarchical metric ~ with the following properties.
(i) ~ is consistent with d, and
(ii) ff ~’ is any other hierarchical metric consistent with d, then ~’(P~,Pi) 0. The Minimum
Spanning Tree Problem, on the other hand, is a minimization problem of
a very different flavor: there are now just a~ finite number of possibilities,
for how the minimum might be achieved--rather than a continuum of
possibilities--and we are interested in how to perform the computation
without having to exhaust this (huge) finite number of possibilities.
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One Can ask what happens when these two minimization issues
are brought together, and the following question is an example of this.
Suppose we have a connected graph G = (V, E). Each edge e now has a timevarying
edge cost given by a function fe :R-+R. Thus, at time t, it has cost
re(t). We’l! assume that all these functions are positive over their entire
range. Observe that the set of edges constituting the minimum spanning
tree of G may change over time. Also, of course, the cost of the minimum
spanning tree of G becomes a function of the time t; we’ll denote this
function ca(t). A natural problem then becomes: find a value of t at which
cG(t) is minimized.
Suppose each function fe is a polynomial of degree 2: re(t) =aet z +
bet + Ce, where ae > 0. Give an algorithm that takes the graph G and the
values {(ae, be, ce) : e ~ E} and returns a value of the time t at which the
minimum spanning tree has minimum cost. Your algorithm should run
in time polynomial in the number of nodes and edges of the graph G. You
may assume that arithmetic operations on the numbers {(ae, be, q)} can
be done in constant time per operation.
In trying to understand the combinatorial StlXlcture of spanning trees,
we can consider the space of all possible spanning trees of a given graph
and study the properties of this space. This is a strategy that has been
applied to many similar problems as well.
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Exercises
Here is one way to do this. Let G be a connected graph, and T and T’
two different spanning trees of G.. We say that T and T’ are neighbors if
T contains exactly one edge that is not in T’, and T"contains exactly one
edge that is not in T.
Now, from any graph G, we can build a (large) graph 9~ as follows.
The nodes of 9~ are the spanning trees of G, and there is an edge between
two nodes of 9C if the corresponding spanning trees are neighbors.
Is it true that, for any connected graph G, the resulting graph ~
is connected? Give a proof that ~K is always connected, or provide an
example (with explanation) of a connected graph G for which % is not
connected.
Suppose you’re a consultant for the networking company CluNet, and
they have the following problem. The network that they’re currently
working on is modeled by a connected graph G = (V, E) with n nodes.
Each edge e is a fiber-optic cable that is owned by one of two companies-creatively
named X and Y--and leased to CluNet.
Their plan is to choose a spanning tree T of G and upgrade the links
corresponding to the edges of T. Their business relations people have
already concluded an agreement with companies X and Y stipulating a
number k so that in the tree T that is chosen, k of the edges will be owned
by X and n - k - 1 of the edges will be owned by Y.
CluNet management now faces the following problem. It is not at all
clear to them whether there even exists a spanning tree T meeting these
conditions, or how to find one if it exists. So this is the problem they put
to you: Give a polynomial-time algorithm that takes G, with each edge
labeled X or Y, and either (i) returns a spanning tree with e~xactly k edges
labeled X, or (ii) reports correctly that no such tree exists.
Given a list of n natural numbers all, d 2 ..... tin, show how to decide
in polynomial time whether there exists an undirected graph G = (V, E)
whose node degrees are precisely the numbers d~, d2 ..... dn. (That is, ff
V = {Ul, v2 ..... vn}, then the degree of u~ should be exactly d v) G should not
contain multiple edges between the same pair of nodes, or "!oop" edges
with both endpoints equal to the same node.
Let G = (V, E) be a graph with n nodes in which each pair of nodes is
joined by an edge. There is a positive weight w~i on each edge (i,]); and
we will assume these weights satisfy the triangle inequality tv~k